Theorem lsmdisjr

Description: Disjointness from a subgroup sum. (Contributed by Mario Carneiro, 21-Apr-2016)

	Ref	Expression
Hypotheses	lsmcntz.p	|- .(+) = ( LSSum ` G )
	lsmcntz.s	|- ( ph -> S e. ( SubGrp ` G ) )
	lsmcntz.t	|- ( ph -> T e. ( SubGrp ` G ) )
	lsmcntz.u	|- ( ph -> U e. ( SubGrp ` G ) )
	lsmdisj.o	|- .0. = ( 0g ` G )
	lsmdisjr.i	|- ( ph -> ( S i^i ( T .(+) U ) ) = { .0. } )
Assertion	lsmdisjr	|- ( ph -> ( ( S i^i T ) = { .0. } /\ ( S i^i U ) = { .0. } ) )

Proof

Step	Hyp	Ref	Expression
1		lsmcntz.p	|- .(+) = ( LSSum ` G )
2		lsmcntz.s	|- ( ph -> S e. ( SubGrp ` G ) )
3		lsmcntz.t	|- ( ph -> T e. ( SubGrp ` G ) )
4		lsmcntz.u	|- ( ph -> U e. ( SubGrp ` G ) )
5		lsmdisj.o	|- .0. = ( 0g ` G )
6		lsmdisjr.i	|- ( ph -> ( S i^i ( T .(+) U ) ) = { .0. } )
7		incom	|- ( S i^i ( T .(+) U ) ) = ( ( T .(+) U ) i^i S )
8	7 6	eqtr3id	|- ( ph -> ( ( T .(+) U ) i^i S ) = { .0. } )
9	1 3 4 2 5 8	lsmdisj	|- ( ph -> ( ( T i^i S ) = { .0. } /\ ( U i^i S ) = { .0. } ) )
10		incom	|- ( T i^i S ) = ( S i^i T )
11	10	eqeq1i	|- ( ( T i^i S ) = { .0. } <-> ( S i^i T ) = { .0. } )
12		incom	|- ( U i^i S ) = ( S i^i U )
13	12	eqeq1i	|- ( ( U i^i S ) = { .0. } <-> ( S i^i U ) = { .0. } )
14	11 13	anbi12i	|- ( ( ( T i^i S ) = { .0. } /\ ( U i^i S ) = { .0. } ) <-> ( ( S i^i T ) = { .0. } /\ ( S i^i U ) = { .0. } ) )
15	9 14	sylib	|- ( ph -> ( ( S i^i T ) = { .0. } /\ ( S i^i U ) = { .0. } ) )